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The given equation is:

\[
16^{2x - 3} = 4^{x + 2}
\]

We can solve this by rewriting both sides in terms of the same base. Notice that:

\[
16 = 4^2
\]

Therefore, the equation becomes:

\[
(4^2)^{2x - 3} = 4^{x + 2}
\]

Using the power rule \((a^m)^n = a^{m \cdot n}\), we get:

\[
4^{2(2x - 3)} = 4^{x + 2}
\]

Simplify the exponents:

\[
4^{4x - 6} = 4^{x + 2}
\]

Now, since the bases are the same, we can equate the exponents:

\[
4x - 6 = x + 2
\]

Solve for \(x\):

\[
4x - x = 2 + 6
\]

\[
3x = 8
\]

\[
x = \frac{8}{3}
\]

Thus, the solution is:

\[
x = \frac{8}{3}
\]

Would you like further clarification on any step? Here are 5 related questions you might want to explore:

1. How can you apply logarithms to solve exponential equations like this one?
2. What is the significance of matching bases when solving exponential equations?
3. How do you solve an exponential equation if the bases cannot be made the same?
4. What are other methods for solving exponential equations?
5. How can you verify the solution to this problem?

**Tip:** When solving exponential equations, always check if both sides can be rewritten with the same base—it simplifies the process significantly.

Learn how to solve the exponential equation 16^{2x - 3} = 4^{x + 2} with a detailed step-by-step solution. This problem covers key algebraic concepts, including rewriting exponents with the same base and applying properties of exponents. Suitable for students in Grades 9-12.

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